Primel Metrology Wiki
(This wiki is under construction. Best viewed in Full Site.) Welcome to the Primel Metrology Wiki The Primel Metrology is a new coherent dozenal-metric system of measurement based on certain fundamental realities of human life on Earth, including the mean solar day, the acceleration due Earth's gravity, the maximal density of water, the specific heat capacity of water, and other properties, with all units derived via 1:1 relationships without extraneous factors. Dozenal (base twelve) arithmetic Primel is a measurement system thoroughly grounded in duodecimal or "dozenal" base arithmetic. Given the high factorability of the number twelve compared to ten, dozenal is arguably a more convenient base for human use than decimal. It is for that very reason that so many historical systems of measure naturally incorporated factors of twelve. But what such systems lacked was a systematic design. Primel can be characterized as a "dozenal-metric" metrology, similar to the Tim-Grafut-Maz (TGM) metrology devised by Tom Pendlebury. Like TGM, Primel systematizes its units around powers of twelve to the same degree that the Metric system (now known as the International System of Units, or SI) systematizes its units around powers of ten. This wiki compares dozenal quantities of Primel units of measure with many decimal quantities of conventional SI and United States Customary (USC) units of measure. To avoid confusion, this wiki will explicitly annotate the base of every number longer than a single digit. It will use the standard mathematical convention, placing the base annotation into a right subscript. However, instead of expressing the base itself as a number in decimal, a subscript "d" will indicate decimal base, and a subscript "z" will indicate dozenal base: For instance, Ӿ = 10d, Ɛ = 11d, 10z = 12d, 100z = 144d, etc. The English language is fairly well-equipped with a nomenclature for dozenal arothmetic, having well-established terms for the first and second powers of twelve: one dozen and one gross. The situation is not so good at the third power, traditionally referred by the rather awkward phrase "a great gross". With each higher power simply prefixing another "great", the result quickly gets unwieldy. Instead of this, this wiki adopts the expedient of co-opting "galore", an English word of Irish origin, meaning "in abundance". This word is unusual in being a post-positive adverb; in other words, unlike other English adjectives or adverbs, it always follows the word it modifies, and never precedes it. This means that we have an opportunity to ascribe another meaning to it if we position it pre-positively, without necessarily interfering with its customary meaning when post-positioned. Therefore, Primel proposed to use one galore for the third power of twelve, one dozen galore for the fourth power, and one gross galore for the fifth. For higher powers still, Primel concatenates "galore" with a numeric prefix derived from Classical Latin or Greek, indicating a certain power of one galore. Thus rather than coining an analog for "million", Primel proposes calling the sixth power of twelve one bigalore, the seventh power of twelve one dozen bigalore, the eighth one gross bigalore, the ninth one trigalore, and so forth. Systematic Dozenal Nomenclature Systematic Dozenal Nomenclature (SDN) provides a set of dozenal power prefixes (usable with any metrology) that are analogous to SI's decimal scaling prefixes. However, SDN does not require an international committee to generate higher and higher order prefixes. Instead, it derives its prefixes systematically from a set of twelve familiar Greek and Latin numeric roots, each directly expressing the exponent in a power of twelve. These roots are identical to those used by the International Union of Pure and Applied Chemistry (IUPAC) to construct Systematic Element Names, extended to support dozenal base with the addition of two roots, dec and lev, to represent ten and eleven as single dozenal digits (Ӿ and Ɛ). IUPAC chose its roots carefully so that they all begin with unique letters, making them amenable to single-character abbreviations. The dozenal additions maintain this uniqueness. SDN also accommodates existing combination forms of these numeric roots, keeping intact their pre-existing meanings as simple multipliers. To avoid clashing with these forms, the power prefixes are derived by appending distinct syllables onto the roots: '-qua' for positive powers, '-cia' for negative powers. The following table lists the roots, the existing multiplier prefixes, and the corresponding power prefixes: Two styles of abbreviations are shown for each prefix. Both combine the initial of a digit root with a special symbol representing the glue syllable for the particular prefix. The first abbreviation, set in san-serif font, represents the glue syllable using a special symbols available in Unicode; this should be the preferred choice for typesetting. The second abbreviation, set in monospaced font, represent the glue syllable using a pure ASCII alternative; this can be used as a substitutes in disadvantaged environments that do not support Unicode. When a multiplier contains multiple digits, or when the exponent of a power contains multiple digits, SDN expresses these by concatenating digit roots. This relies on exactly the same place-value arithmetic principle that Hindu-Arabic numerals employ. The final digit is then terminated as either a multiplier or power form; this determines the form for the whole string. So for example, the following table shows the multiplier and power prefixes corresponding to the next 3 dozenal values beyond eleven: Because the power prefixes are distinct from the multiplier forms, both can be freely combined without ambiguity, to create an analog of scientific notation. So for instance, bihexpenta- (bhp•) represents 265z, bihexa-'pentqua'- (bh•p↑) represents 26×105z, bina-'hexpentqua'- (b•hp↑) represents 2×1065z, and bihexpentqua- (bhp↑) represents 10265z. To completely represent scientific notation, we need one additional lexical element, to represent a fraction point. SDN uses the syllable dot for this purpose (abbreviated with the usual period). Hence, bihexa-'pentqua'- (bh•p↑), representing 26×105z, can also be expressed as bidothexa-'hexqua'- (b.h•h↑), representing 2.6×106z. A multiplier digit is only required to the right of dot. If there is no digit to the left, it is assumed to be nil (0). So for example, dothexa- (.h•) represents 0.6z, i.e., a half. SDN also provides a set of prefixes representing the reciprocals of whole numbers, using the marking suffix '-infra' (abbreviated with a backslash \ ). This derives directly from the Latin word infra, which means "below" or "under". The sense is that the preceding digits are being placed under the horizontal line of a fraction. So, for example, a fifth can be expressed as pentinfra- (p\); a seventh as septinfra- (s\); and so forth. These can be freely combined with the ordinary multipliers to express any rational number. Thus, 5/7 could be expressed as penta-'septinfra'-. (p•s\). Interestingly, these reciprocal prefixes act as distinct multiplicative factors, so there is no order dependency like there would be with an actual division operator. Hence 5/7 could be expressed equivalently as septinfra-'penta'- (s\p•). This will not be confused with 7/5, because that would be expressed as septa-'pentinfra'- (s•p\), or pentinfra-'septa'- (p\s•). The syllable per- can be combined with powers of dozen to provide dozenal analogs of decimal percent (%) and permille (‰). Dozenalists have often expressed these as "pergross" and "per great-gross". However, in SDN, these can be pronounced perbiqua and pertriqua. To some extent these are redundant with bicia and tricia, but it is often helpful to think of a fraction as a number of parts from a group. So for example, a ratio of 1/3 could be expressed as 40%z ("four dozen perbiqua") or even 400‰z ("four gross pertriqua"). This could be extended to any power of dozen, so for instance analogs for "parts per million", "parts per billion", "parts per trillion", etc., could be expressed as perhexqua, perennqua, perunnilqua, etc. Quantitels Quantitelsare systematic names for units of measure derived directly from the names of the physical quantities they measure, plus the ending -'el', which signifies "element", as in the familiar word "pixel" = "picture element". For example, lengthel is the quantitel for a unit of length. When a quantity has multiple synonyms (e.g. "work" and "energy"), it can have multiple synonymous quantitels (e.g. workel and energel); such synonyms can be used interchangeably. Quantitels are generic and can be used across potentially many metrologies. This particular metrology is named "Primel"' '''because it is the first (i.e., prime) system to use such unit names. "'Primel'" can be used as a disambiguating adjective to distinguish Primel quantitels from those of other metrologies (e.g. '''Primel lengthel'), but this is optional and may be omitted when the discussion is exclusively about Primel units. The prime character ( ′''' ) serves as an abbreviation for this prefix, marking every Primel unit as such (e.g. '''′lengthel). It may be left silent, or pronounced "prime" or "Primel" as needed. Colloquial Names In addition to systematic quantitel names, Primel proposes "colloquial" names, or "nicknames", for some of the base units, as well as for some useful multiples and dozenal powers of the base units. Some of these proposed nicknames will be purely fanciful. For instance, because the ′timel is rather short and fleeting, Primel proposes the colloquial name ′jiff for this. Because the unqua′timel is about the time to blink an eye, Primel proposes nicknaming this the ′twinkling. Because the ′lengthel is about the size of a bit of food pinched between thumb and forefinger, Primel proposes nicknaming it the ′morsel-length. And so forth. But in many cases a nickname will be proposed because a unit closely approximates a customary or SI unit. For instance, because the trina′lengthel approximates the customary inch (which in other languages is called a "thumb", e.g. Latin pollex), Primel proposes to nickname this the ′pollical-length. Because the unqua′lengthel approximates the customary "hand" measure (Latin manus), Primel proposes to nickname this the ′manipular-length. Because the trina-unqua′lengthel approximates the customary foot (Greek pous, podis), Primel proposes to nickname this the ′podial-length. And so forth. Primel colloquial names often follow a pattern that concatenates the thematic "prime" prefix, plus a "colloquial" adjective (often of Classical origin and usually ending in '-al' or '-ar'), plus the plain English word for the physical quantity being measured. For instance, the colloquial adjective manipular, plus the physical quantity length, yields the nickname ′manipular-length for the unqua′lengthel. Primel leverages this pattern to name related units for derivative physical quantities, by reusing these colloquial adjectives with different physical quantity names. For instance, the biqua′areanel, an area of one square ′manipular-length, is nicknamed a ′manipular-area. The triqua′volumel, a volume of one cubic ′manipular-length, is nicknamed the ′manipular-volume. The triqua′massel, a mass of one ′manipular-volume of water, is nicknamed the ′manipular-mass. The triqua′forcel or triqua′weightel, the force/weight of one ′manipular-mass in Earth's gravity, is nicknamed a ′manipular-force or ′manipular-weight. The quadqua′energiel or quadqua′workel, the energy or work needed to lift a ′manipular-mass by one ′manipular-length against Earth's gravity, is nicknamed a ′manipular-energy or ′manipular-work. The unqua′timel, the time to traverse one ′manipular-length at one ′velocitel, is nicknamed the ′manipular-time. The triqua′powerel, a rate of power which applies one ′manipular-energy per each ′manipular-time, is nicknamed a ′manipular-power. The unqua′pressurel, a pressure which applies one ′manipular-weight per each ′manipular-area, is nicknamed a ′manipular-pressure. And so forth. Primel's length units have a robust set of colloquial names running up and down the magnitude scale, from the microscopic to the macroscopic, associating these lengths with certain objects that exist at those scales. Because of this, and because of the above naming pattern, Primel can construct colloquial names for many derivative units at all these scales. 1:1 Coherence Primel endeavors, where feasible, to relate the base units for different physical quantites using simple 1:1 ratios, without extraneous factors. (TGM also adheres to this principle.) Primel starts deriving its units by considering time first. The Day Primel uses a pure dozenal fraction of the mean solar day, namely the hexciaday (10-6z day), as the base unit of time (the ′timel), equivalent to 50/1728d (0.042z) seconds (nickname: ′jiff). (This differs from TGM, which starts with a pure dozenal fraction of the hour instead.) Because of 1:1 coherence, this choice affects the sizes of all other base units in the metrology. It turns out that this yields many conveniently-sized units, either in the base units themselves, or when scaled using simple whole number multiples and/or pure dozenal powers (in the form of SDN prefixes). Earth's Gravity For its unit of acceleration, the ′accelerel, Primel uses a value for net gravitational acceleration experienced on Earth's surface. This varies over an appreciable range, based on a number of factors, but chiefly latitude, due to the effect of centrifugal force produced by Earth's rotation about its axis. Within this range, Primel chooses as its standard a value of exactly 9.79651584d m/s2, or exactly 32.1408d ft/s2. This choice allows for exact conversions between Primel units and both SI and USC units, not only for acceleration, but also for velocity and length, which Primel derives from the ′accelerel via the principle of 1:1 correspondence. The ′accelerel is somewhat lower than the SI standard for Earth's gravity, which is often described as an "average" of Earth's gravity. However, the SI standard appears to be a 19thd-Century estimate of gravity at median latitude, i.e. 45°d or 16%⊙z (16z "turnlets", an eighth of a turn, or 1 octant, of latitude). Latitudes cover progressively more surface area toward the equator, so the actual average gravity when integrated over Earth's surface area is somewhat lower. The Primel standard approximates this much more closely than the SI standard. The ′velocitel is the velocity a body achieves after falling for 1 ′timel under 1 ′accelerel of acceleration. This turns out to be remarkably close to 1 kilometer per hour (it is exactly 1.0204704d km/h), or 1 foot per second (it is exactly 0.93d fps). The ′lengthel is the distance traversed by an object moving at a constant 1 ′velocitel for 1 ′timel. It is exactly 31/96d(0.322916d) inches, or exactly 8.202083d mm. Primel nicknames this the ′morsel-length. This results in a trina′lengthel (nickname: ′pollical-length) of exactly 0.96875d inch, or exactly 24.60625d mm, which closely approximates the USC inch; an unqua′lengthel (nickname: ′manipular-length) of exactly 3.875d inch, or exactly 98.425d mm, which closely approximates both the 4-inch USC hand measure as well as the SI decimeter; a trinan-unqua′lengthel (nickname: ′podial-length) of exactly 11.625d inches or 0.295275d m, which approximates the USC foot; and a biqua′lengthel (nickname ′ulnaral-length) of exactly 46.5d inch or 1.1811d m, which approximates the traditional 45d-inch English ell. From the ′lengthel in turn, Primel derives units of area and volume, the ′areanel (1 square ′lengthel) and the ′volumel (1 cubic ′lengthel). The ′volumel is approximately 0.5518d ml. The triqua′volumel (nickname, ′manipular-volume) comes remarkably close to a customary quart or an SI liter. (It is about 1.00754d quarts or 0.95349d liters.) Density of Water Primel uses the maximal density of water as its base unit of density, the ′densitel. This leads to a unit of mass, the ′massel, of about 0.55d grams (nickname: ′morsel-mass); with the triqua′massel (nickname: ′manipular-mass) being just under 1 kilogram and just over 2 pounds. Because Earth's gravity is the unit of acceleration (1 ′accelerel = 1 ′gravity), whatever the mass of anything is in ′massels, the force of its weight in ′forcels (or ′weightels) will be numerically the same (approximately, on Earth). So we could easily speak of the mass of something in ′morsel-masses or ′manipular-masses, and its weight in ′morsel-weights or ′manipular-weights, using (approximately) the same magnitudes. (Contrast this with the situation in SI, with kilograms of mass versus newtons of weight, with the factor of 9.80665d m/s2 in between.) Units for energy (the ′energiel or ′workel), power (the ′powerel), pressure (the ′pressurel), and the rest of Newtonian mechanics, are derived in straightforward fashion. Specific Heat Capacity of Water To relate the phenomenon of heat and thermodynamics to the foundations of mechanics, Primel starts with a base unit for specific heat capacity (the ′massic-heatcapacitel). This is set to a representative value for the specific heat capacity of water, within its natural range. This leads to a unit for temperature (the ′temperaturel) defined as the increase in temperature induced in a 1 ′massel sample of water by applying 1 ′workel of heat energy. Since this turns out to be a very tiny amount of temperature, a dozenal power of this, the quadqua′temperaturel, comes out to about 0.4d kelvins, and thus is convenient for everyday use. To be precise, the quadqua′temperaturel is defined as exactly 5/7 of a Fahrenheit degree (about 0.7143d°F), or exactly 25/63d (21/53z) of a Celsius degree (about 0.3968d°C), such that the range from water's melting point to its boiling point is exactly 190z (252d) quadqua′temperaturels, a round multiple of twelve. The ′massic-heatcapacitel has been set to about 4198.76d J/K/kg specifically to yield this result; it is slightly above the theoretical average specithermacity of water over its liquid phase (4190d J/K/kg), but slightly less than the so-called "dietary kilocalorie" (4200d J/K/kg). This unit is nicknamed the ′stadigrade, because of the equivalence of heat energy to mechanical work: The heat required to raise the temperature of a body of water by one ′stadigrade is the same amount of energy as the mechanical work necessary to lift the same body of water one ′stadium (1 quadqua′lengthel) upward against Earth's gravity. Primel provides three temperature scales based on the ′stadigrade, but with different choices for a zero point: an Absolute scale zeroed at absolute zero, analogous to the Kelvin and Rankine scales; a Crystallic scale zeroed on the freezing point of water, somewhat resembling the Celsius scale; and a Familiar scale zeroed 40z ′stadigrades below freezing, bearing remarkable similarities to the Fahrenheit scale. Impedance of Free Space For the moment, the table below shows what Primel electrical units would look like if the impedance of free space were used as the base unit of resistance/reactance/impedance, with all other electrical units derived from that. However, the author has not yet decided whether to settle on this scheme. While it has a certain elegance and symmetry from a theoretical basis, this would not make the electrical system as "friendly" to interconversion with SI and USC as the mechanical or thermodynamic systems. The author is still investigating whether to go with an approach similar to SI's, based on Ampere's Force Law. Table of Primel Base Units The following table provides an executive summary of the Primel base units for each type of physical quantity. The names in the first column (will) serve as links to the wiki pages covering each type of physical quantity in more detail. Latest activity